## 1. Introduction

All-optical signal processing is essential for optical communication systems transmitting high speed data signals. All-optical wavelength conversion (WC), pulse generation, orthogonal frequency-division multiplexing (OFDM), demultiplexing, regeneration, modulation format conversion are the important signal processing functions. The limiting factors in the existing ultra long haul optical communication systems are spectral efficiency (SE), fiber attenuation, insertion losses, chromatic dispersion, polarization mode dispersion (PMD), and the optical fiber nonlinearity [1]-[4]. Optical communication systems can be divided into two groups. In the first group, an electrical signal modulates the intensity of an optical carrier inside the optical transmitter, the modulated optical signal is transmitted through the optical fiber and converted to the original electrical signal by an optical receiver [1]. Such a scheme is called an intensity modulation with direct detection (IM/DD) [1], [5]. In the second group, information is transmitted by modulation of the optical carrier frequency or phase, and then the modulated optical signal can be linearly down-converted to a baseband electrical signal by heterodyne or homodyne detection [1], [5]. The phase coherence of the optical carrier is essential for the realization of the second group called coherent optical communication systems [1].

Coherent optical communication systems possess the following advantages with respect to IM/DD systems: (i) the shot-noise limited receiver can be achieved with a sufficient local oscillator (LO) power; (ii) the frequency resolution at the intermediate frequency (IF) or baseband stage is high enough in order to separate close wavelength-division multiplexed (WDM) channels in the electric domain; (iii) the phase detection improves the receiver sensitivity compared with IM/DD systems; (iv) the multilevel modulation formats can be introduced into optical communications by using phase modulation [5].

The rapid development of digital communication and digital signal processing (DSP) caused by the necessity of high SE and, at the same time, by the advance of the high-speed electronics required advanced modulation, coding and digital equalization [6]. Recently, coherent optical communications attracted a large interest due to the feasibility of the high SE, the large bandwidth and multilevel modulation formats [5]. Coherent optical systems using multilevel modulation formats can increase SE up to *M* b/s/Hz where *M* is the number of bits per symbol for a given modulation format [5].

Coherent optical OFDM (CO-OFDM) has been recently proposed in order to increase receiver sensitivity, SE, and especially, provide the dispersion compensation at high data transmission rates tending to 100Gb/s [3], [4], [7], [8]. Generally, OFDM is a kind of multicarrier modulation (MCM) in which the data information is carried over many lower rate subcarriers [4]. In OFDM spectra of individual subcarriers overlap, but due to the orthogonality, the subcarriers can be demodulated without interference and without analog filtering for the received subcarrier separation [7]. The signal processing in the OFDM system can be carried out by using the Fast Fourier Transform (FFT)/Inverse Fast Fourier Transform (IFFT) [3], [4], [7]. In CO-OFDM systems the digital signal processing is used in order to mitigate the channel dispersion, nonlinearity and different types of noise.

CO-OFDM system combines the following advantages of coherent detection and OFDM modulation essential for the high-speed optical fiber communication systems: (i) in CO-OFDM systems, chromatic dispersion and PMD can be mitigated; (ii) the SE of CO-OFDM systems is high due to the partial overlapping of OFDM sub-carrier spectra; (iii) linearity in radio frequency (RF)-to-optical (RTO) up-conversion and optical-to-RF (OTR) down-conversion; (iv) the electrical bandwidth requirements for the CO-OFDM transceiver can be greatly reduced by using direct up/down conversion which results in the low cost of the high-speed electronic circuits [3], [4], [8]. Recently, all-optical FFT scheme enabling Tbit/s real-time signal processing has been proposed [9]. The method based on only passive optical components realizes the highest speed signal processing without the power consumption where electronics cannot be used. This approach combines the advantages of the electronic high precision processing of the low bit rates and the optical processing of high bit rates [9].

However, OFDM is characterized by the inter-symbol-interference (ISI) and inter-carrier-interference (ICI) caused by a large number of subcarriers [4]. In the RF systems ISI is mainly due to multipath channel delay spread [10], [11] and ICI is mainly due to the carrier frequency offset [12]. In the case of CO-OFDM, ISI and ICI are caused by channel chromatic dispersion and PMD [3], [4]. A so-called cyclic prefix (CP), i.e. the cyclic extension of the OFDM waveform into the guard interval (GI)

The need for CP can be avoided if the wavelet packet transform (WPT) is used in CO-OFDM systems instead of discrete Fourier Transform (DFT) and inverse DFT (IDFT) [13]. The sinusoidal functions are infinitely long in the time domain while wavelets have finite length being localized in time and in frequency domains [13]. Wavelet signal analysis can be a base for an effective computational algorithm which is faster and simpler than FFT [14]. Wavelets have been used in optical communications for time-frequency multiplexing and ultrafast image transmission [14]. A signal may be expanded in an orthogonal set of wavelet packets (WPs) as the basis functions, each channel occupies a wavelet packet (WP), and IDWPT/ DWPT are used at the transmitter and receiver, respectively [13].

In this chapter, we consider the CO-OFDM based on WPT and its influence on the optical communication network performance. The chapter is constructed as follows. In Section 2, we review the coherent optical communication systems. In Section 3, we discuss high SE CO-OFDM system. In Section 4, we discuss the OFDM based on WPT and present the original results for the WPT-OFDM system performance. In Section 5, we present the original results concerning the simulations of the structure and operation mode of the novel passive components for all-optical signal processing based on Si-on-insulator (SOI) structure, and a novel hierarchical architecture of the 1Tb/s transmission system based on WPT-OFDM [15]. In Section 6, conclusions are presented.

## 2. Coherent optical communication systems

The number of publications concerning the coherent optical communications is enormous. In this section, we can only relate to a limited number of the fundamental works concerning the concept of the coherent detection and the modulation formats used in digital communication system since these topics are related to high SE CO-OFDM systems.

The most advanced detection method is coherent detection based on the recovery of the full electric field containing both amplitude and phase information [16]. The concept of the coherent detection is to combine in a receiver the modulated optical signal with a continuous wave (CW) optical field generated by a narrow linewidth laser, or local oscillator (LO) before the photodetector (PD) [1]. Coherent detection requires the carrier synchronization with respect to LO that serves as an absolute phase reference [16]. For this purpose, optical systems can use two types of the phase-locked loops (PLLs): (i) an optical PLL (OPLL) that synchronizes the frequency and phase of the LO laser with the transmitter laser; (ii) an electrical PLL where down-conversion with a free-running LO laser is followed by a second-stage demodulation by an electrical voltage controlled oscillator (VCO) with the synchronized frequency and phase [16].

The basic component of coherent optical systems is a coherent optical receiver [1], [5]. Its block diagram is shown in Figure 1 [5].

The electric fields

where

The output photocurrents

where

Eq. (4) demonstrates the main advantage of the coherent detection as compared to the direct detection. The photocurrent

Similarly, the increase of the average electrical power up to 20dB can occur in the case of the homodyne detection. If additionally, the LO phase is locked to the signal phase so that

The phase difference

The coherent detection allows the greatest flexibility in modulation formats, since information can be encoded by modulating the amplitude, the phase, or the frequency of an optical carrier as it is seen from equations (1)-(6) or in both in-phase (I) and quadrature (Q) components of the carrier [1], [16]. In the case of the digital communication systems, these methods correspond to three modulation formats: amplitude-shift keying (ASK), phase--shift keying (PSK), and frequency-shift keying (FSK) [1]. The increased performance, speed, and reliability, and the reduced size and cost of integrated circuits permit to use DSP for the information recovery from the baseband signal [5]. Typically, the M-ary PSK modulation is used in SE high-speed CO-OFDM systems such as quaternary PSK (QPSK) (M=4), 8-PSK (M=8), as well as quadrature amplitude modulation (QAM) such as 4-QAM, 16-QAM, 64-QAM in single or dual polarization [4], [5]. The digital coherent receiver linearly detects incoming signal including phase and polarization diversities and converts this information to digital data by using ADCs while the digital information is processed by DSP circuits [5].

## 3. High-speed and high SE CO-OFDM system

In this section we present a brief review of the operation principle and architecture of CO-OFDM system. Detailed analysis of CO-OFDM in optical communication systems may be found in the book [4].

A generic optical OFDM system consists of five functional blocks: the RF OFDM transmitter, the RTO up-converter, the optical channel, the OTR down-converter, and the RF OFDM receiver [3], [4]. In such a system the following chain of events occurs [3]. The input data bits are mapped onto corresponding information symbols of the subcarriers within one OFDM symbol. The digital time domain signal is obtained by using IDFT. It is inserted with the GI Δ_{G} in order to prevent ISI caused by channel dispersion and converted into the real time waveform through DAC [3]. The baseband signal is up-converted to an appropriate RF band with an IQ mixer/modulator. A linear RTO up-conversion can be achieved by using Mach-Zehnder modulator (MZM) [3]. MZM is mainly used for the bit rates of 40GB/s and higher due to its high modulation performance and the possibility of independent modulation of the electric field intensity and phase [6]. At the receiver, the OFDM signal is down-converted to baseband with an IQ demodulator, sampled by an ADC and demodulated by DFT and baseband DSP to recover the data [3]. A linear OTR down-conversion is provided by a coherent detection described in section 2. The high performance of the CO-OFDM transmission systems has been shown both theoretically and experimentally [8], [17]. A single-channel 1Tb/s CO-OFDM signal consisting of 4104 spectrally-overlapped subcarriers with SE of 3.3bit/s/Hz has been generated, transmitted over 600km standard single mode fiber (SSMF) without amplification and dispersion compensation, and successfully received [17]. However, CO-OFDM system is extremely sensitive to nonlinearity and channel dispersion. The dispersion mitigation with the dispersion compensation fiber (DCF) results in the additional noise and nonlinear effects decreasing the system performance [8].

Consider now the analytical expressions describing the CO-OFDM signals. The MCM transmitted signal

where

The detected information signal

where

The OFDM technique permits to use the overlapped signals under the condition that they are orthogonal [7]. The orthogonality condition for any two subcarriers

Substituting expression (8) into condition (10) we obtain.

The left-hand side of (11) vanishes when

Then, the two subcarriers

In the high speed CO-OFDM systems the problem of ISI and ICI caused by the channel dispersion is critical. ISI is caused by the interference between "slow" and "fast" subcarriers. ICI is due to the breaking of the orthogonality condition (12) for the subcarriers [3]. In order to prevent ISI and ICI, CP was proposed that is realized by cyclic extension of the OFDM waveform into GI [3]. The waveform in GI is essentially an identical copy of that in the DFT window [3]. The condition for ISI-free OFDM transmission requires that the dispersive channel time delay spread

SE is defined as the ratio of net per-channel information data rate B to WDM channel spacing

where

## 4. WPT based CO-OFDM

WPT can be used in CO-OFDM instead of the IDFT/DFT since it improves the system performance, and in particular, mitigates the channel chromatic dispersion without CP [13]. In this section we briefly discuss the main features of WPT and its applications to CO-OFDM. The theory and applications of continuous wavelet transform (CWT) and discrete WT (DWT) can be found in a large number of books and articles (see, for example, [13], [14], [19]-[22] and references therein).

CWTwhere the real numbers *a* and *a* and

where m,n∈Z, Z is the set of all integers, and the constants

Here a subspace V₀∈L²(R), L²(R) is a the linear vector space of square integrable functions,

where h[k] and g[k] are the coefficients of two half-band (HB) quadrature mirror filters (QMFs) described by the following functions H(ω) and G(ω)

and Δτ is the inverse of their FSR. The functions H(ω) and G(ω) (18) satisfy the conditions [14], [22].

The evaluation of the discrete wavelet coefficients is equivalent to filtering the signal s(t) by a cascade of mutually orthogonal bandpass filters [21]. An optical HB filter can be realized by using Mach-Zehnder interferometers (MZIs) [14], [22].

The DWT decomposition procedure is described by the following recursive expressions for the scaling and detail coefficients

where

In the DWT case only the scaling coefficients

where

where

WPTs can provide orthogonality between OFDM subcarriers similarly to DFT, and consequently, DWPT can replace DFT in the CO-OFDM system [13]. The all-optical WPT based CO-OFDM (WPDM) system has been proposed where the digital sequences are encoded by a set of orthogonal waveforms [13], [14]. The system performance is improved due to the orthogonal properties (25) of the wavelet atoms (22)–(24) and their overlapping in time and frequency [13], [14]. Each optical pulse is transformed into the corresponding wavelet atom function at the device output under the conditions that the input bit duration

Consider the computational complexity

where

The performance of a digital optical communication systems is characterized by the bit error rate (BER) defined as the average probability of incorrect bit identification [1]. The simulations of the BER for WPT-OFDM and FFT based OFDM have been carried out using different wavelets, optical SNR of 20dB, chromatic dispersion parameter of 17ps/(nm⋅km), and forward error correction code (FEC) threshold of 10⁻³ [13]. The results show the chromatic dispersion tolerance of 5600 ps/nm and the longest distance of 330km for SSMF for the Johnston64 (E) wavelet [13].

We have carried out the numerical simulations of BER dependence on the transmission distance in the single polarization regime for the WPT-OFDM system without CP, with GI length of 5% of the symbol interval, and for generic IDFT/DFT systems with values of CP length from 5% up to 30% of the symbol interval. We used the single-polarization signal with the optical carrier frequency

The BER dependence on the distance for the Haar WPT-OFDM and FFT CO-OFDM with different CPs is shown in Figure 2.

The results clearly show that WPT-OFDM provides the efficient transmission up to 500km without CP with 5% GI, while the FFT based CO-OFDM may achieve the same distance with the CP length of 25% of the symbol interval which substantially reduces SE of the communication system.

## 5. Passive Si-photonic components for all-optical signal processing

In this section we discuss the implementation of passive WPT-OFDM system components based on the Si photonics and a novel hierarchical architecture of the 1Tb/s WPT-OFDM transmission system that can be realized by using these components.

### 5.1. SOI optical components

The practical implementation of all-optical signal processing would require some extent of device integration. Much effort is dedicated over the last two decades to the development of photonic integrated circuits (PICs), which bring together multiple discrete devices on a single substrate. Integration helps to minimize the losses associated with the coupling of light in and out of devices, to enhance functionality, and to reduce cost and footprint. Numerous material platforms are prevalent in PICs, such as LiNbO₃, GaAs, InP and SiO₂. Among those platforms, the SOI wafer structure stands out as an advantageous choice for the realization of passive devices, such as couplers, interferometers, arrayed-waveguide gratings etc [24], [25]. Silicon is a low-cost material with an excellent crystalline quality, high thermal conductivity and high optical damage threshold. It is transparent over a broad range of wavelengths of 1.1-7 μm, including the telecommunication wavelengths. The silica SiO₂ lower cladding of SOI wafers provides a large contrast in refractive index with respect to silicon, which allows for the tight confinement of light into sub-micron structures. The fabrication of photonic devices in SOI can benefit from the advanced manufacturing technology of electronic integrated circuits. Silicon photonic devices may lead to a true merger of optics alongside electronics in unified devices. The realization of modulation of light on the silicon material platform is more challenging. The concentration of free charges in silicon changes the real and imaginary parts of the refractive index, and this effect in pure silicon is more strongly pronounced than the Pockels effect, the Kerr effect and the Franz-Keldysh effect [26]. Most of the fast modulators integrated on Si are based on free-carrier concentration variations [27]. Optical modulation using SiGe/Si and all-silicon phase shifters based on carrier depletion has been investigated theoretically and experimentally [27]. An all-silicon phase-shifter based on carrier depletion in a doped layer inserted into a PIN diode has been demonstrated [28]. SiGe/Si and all-silicon modulators can be integrated in rib waveguides and in MZIs [27]. Another modulation technique for SOI optical devices is based on the thermo-optic effect, in which the refractive index n of silicon is varied by applying heat to the material [24]. The thermo-optic coefficient in silicon is given by

### 5.2. Example of SOI MZM for all-optical signal processing

In this section we present an example of the SOI based MZI which can realize the WPT operation. The most basic family of wavelet shapes is the Haar transform, proposed initially by Alfred Haar in 1910 [19]. The Haar wavelet and scaling functions

Note that the equivalent definitions

The inverse operation recovers the original signal from its decomposition coefficients. Its scheme is shown in Figure 4. Here S[n] is the output signal, g[n] and h[n] are the discrete HPF and LPF impulse responses, c[n] and d[n] are the approximation and detail coefficients respectively mentioned above.

The realization of Haar wavelet packet decomposition (WPD) transform and the corresponding inverse wavelet packet decomposition (IWPD) in an optical integrated circuit was theoretically suggested by Gabriella Cincotti and co-workers [14], [22], [29], [30]. The method is based on the following MZI delay line architecture shown in Figure 5.

The IWPD is represented by the optical field

Expression (29) shows that a single MZI could provide the sum and the difference of its two input fields, in series, in one of its output ports. The operation is equivalent to the LPF and HPF operation of the inverse Haar IWPD. Similarly, the same MZI can generate the sum of successive values in one of its input ports at one output

The latter configuration described by expression (30) can realize the Haar WPD. Hence, MZIs can function as a basic building block of a discrete Haar WPD and IWPD. As can be seen in equations (29) and (30), the MZI realization of the WPD includes an additional relative phase shift of 90° in between the two inputs/outputs, which is not part of the Haar formalism. This additional phase must be compensated for. Furthermore, the optical path lengths connecting between cascaded MZIs cannot be controlled at the fabrication stage to a sub-wavelength precision. Hence, metallic resistors must be deposited in proximity to the waveguides [31], [32]. The driving of currents through the resistors would locally heat the nearby silicon structure, and modify its refractive index through the thermo-optic effect mentioned above [24]. A schematic drawing of a single MZI with the thermo-optic phase-shifters is shown in Figure 6.

Three-stage MZI-based photonic integrated circuits for the realization of Haar WPT-OFDM encoding and decoding based on the single MZI stage are shown in Figures 7, 8.

In the Haar WPT-OFDM encoder presented in Figure 7, S₁-S₈ are low rate input data channels, with a seven-bits zero padding. The output is the multiplexed Haar transform signal.

In the all-optical Haar WP Decoder shown in Figure 8, the input signal is constructed from eight data channels, which are recovered individually at the eight outputs. The output channels must be down sampled by factor of 8.

The all-optical WP encoder calculates the Haar IWPD of eight coefficients, incoming from eight parallel input values. The reconstructed signal appears in series at the output of the circuit. Note that padding by seven zeros is necessary between successive bits at each input, so that the transform coefficients of one input parallel word do not overrun those of the next word at the output [9], [33], [34]. The zero padding is the optical-domain equivalent of the up-sampling that is part of a digital IWPD. Similarly, a proper gating is necessary at the each of the eight outputs of the decoder circuit, since the original data is only reconstructed at specific time slots within the symbol duration [33]. The remainder of the symbol duration is occupied by noise-like ISI.

A WPT based CO-OFDM communication network, employing the encoding and decoding PICs, is shown in Figure 9. Light from a CW laser diode is split in eight paths. Light in each path is individually modulated by a separate stream of data, which are prepared with the necessary zero padding as described above. The eight channels are multiplexed by the WPT-OFDM PIC. At the other end of the link, each of the eight outputs of the WPT-OFDM decoder PIC is separately gated by an electro-optic switch and detected.

The SOI waveguide is a basic component of the Si photonic systems. We calculated the modal profile of such a waveguide in a single mode regime for each polarization of the optical wave. The SOI waveguide cross-section and the modal profile are shown in Figure 10. The analysis of such waveguides can be carried out only by numerical methods [35]. We used the commercial software modeling (COMSOL). The modal field distribution (Figure 10b) clearly shows the electric field confinement in the waveguide core.

A basic building block of a MZI is a directional coupler. Couplers are realized by bringing two SOI waveguides in close proximity for a certain length Z₀. The degrees of freedom in the design are the length and gap between the SOI waveguide cores. A relatively large gap of the order of magnitude of 300nm is advantageous with respect to fabrication imperfections. COMSOL simulations were used to calculate the coupling coefficient

Consider now the differential delays of the MZIs. As discussed earlier, different stages in the cascaded MZI PIC require different delays. The basic delay unit is T/8, where T is the symbol duration. For a data rate of 2.5 GSymbols/s for each of the eight multiplexed channels, the fundamental delay unit is 50psec, which corresponds to a physical length of about 3.5mm in SOI waveguides. The heat dissipation from aluminium heaters in proximity of the SOI waveguides was simulated, once more using COMSOL. Figure 12 shows the resulting temperature profile. The Al heaters are heated by an external current up to 60°C. Simulation results show that a temperature in the Si core of the SOI waveguide is 40°C compared to 20°C in the unheated regions. This temperature difference stems from the attachment of the back end of the PIC to a 20°C heat sink.

### 5.3. Hierarchical architecture and performance of the WPT based OFDM system

The DSP in CO-OFDM systems is carried out by the algorithms realized with the field programmable gate array (FPGA) and application specific integrated circuit (ASIC) electronic processors. Their computational power is limited with the operation rate of the VLSI electronic elements. For this reason, the electronic "bottleneck" can be eliminated and the system operation rate can be improved if high data rates signal processing is realized by all-optical methods such as all-optical WPT-OFDM.

We proposed a novel hierarchical architecture of the 1Tb/s transmission system based on WPT-OFDM in order to reduce the computational complexity of the DSP algorithms [15]. The hierarchical architecture concept is based on the separation of low bit rate and high bit rate signal channels, unlike the system discussed in Ref. [13]. We used an IDWPT/DWPT system based on the Haar WPT with the wavelet function

The WPT-OFDM hierarchical transmitter and receiver are shown in Figures 13 and 14, respectively.

The high data rate bands are multiplexed using all-optical IDWPT. The transmitter includes IQ modulator. QAM 16, QAM 4 and other multilevel modulation formats can be used for each subband. Subbands are multiplexed in electrical domain also by utilizing the IDWPT. At the receiver side, the multiband signal is demultiplexed into the 8 bands using all-optical

filters and consequently demultiplexed into the subbands in the electrical domain by the DWPT. The performance of the WPT-OFDM communication system based on the hierarchical architecture has been investigated theoretically taking into account the 10% non-ideality of the system devices and the white Gaussian noise model. We used the modulation format QAM 16, 8 level decomposition, the bit rate of 500Gb/s. The 500Gb/s multiple band consists of eight 62.5Gb/s bands. The simulated constellation of the transmitter-receiver "back-to-back" system, i.e. without optical fiber link is shown in Figure 15.

The constellation clearly shows the high performance of the transmitter-receiver system.

## 6. Conclusions

We discussed in this chapter the structure, operation principle and basic properties of the all-optical high SE CO-OFDM systems. The CO-OFDM is the most promising modulation method in the modern optical systems combining the advantages of the coherent detection and OFDM modulation. However, the high data rate transmission is strongly influenced by the optical channel chromatic dispersion and PMD. As a result, the transmission system performance significantly deteriorates due to ISI and ICI. The dispersion influence can be mitigated by an appropriately chosen CP. Unfortunately, a long enough CP would decrease the system SE. The problem can be solved and the necessity of CP may be eliminated if a generic DFFT based CO-OFDM is replaced with a WPT-OFDM since WPs are localized both in time and frequency domains. The simulation results show that WPT-OFDM system provides a 500 km transmission distance at the FEC level of

### Acknowledgement

This work was supported in part by the Israeli Science Foundation (ISF), and by the Chief Scientist Office of the Israeli Ministry of Industry, Trade and Labor within "Tera Santa" consortium.